Subtract the following rational expressions. $\dfrac{8z^4}{8z+20}-\dfrac{2}{z^5}=$
Explanation: We can subtract two rational expressions whose denominators are equal by subtracting the numerators and keeping the denominator the same. [Does this fit with how we subtract rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Note that we can simplify the first expression before we add the two expressions. It is useful to do this, when possible, since then we have simpler expressions to deal with: $\dfrac{8z^4}{8z+20}=\dfrac{\cancel{{(4)}}\cdot2z^4}{\cancel{{(4)}}\cdot(2z+5)}=\dfrac{2z^4}{2z+5}$ Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({2z+5})\cdot({z^5})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{2z^4}{{2z+5}}-\dfrac{2}{{z^5}} \\\\ &=\dfrac{2z^4\cdot({z^5})}{({2z+5})\cdot({z^5})}-\dfrac{2\cdot({2z+5})}{({z^5})\cdot({2z+5})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's subtract! $\begin{aligned} &\phantom{=}\dfrac{2z^4\cdot(z^5)}{(2z+5)\cdot(z^5)}-\dfrac{2\cdot(2z+5)}{(z^5)\cdot(2z+5)} \\\\ &=\dfrac{2z^4\cdot(z^5)-2\cdot(2z+5)}{(2z+5)(z^5)} \\\\ &=\dfrac{2z^9-4z-10}{(2z+5)(z^5)} \end{aligned}$ In conclusion, $\begin{aligned}\dfrac{8z^4}{8z+20}-\dfrac{2}{z^5}&=\dfrac{2z^9-4z-10}{(2z+5)(z^5)}\end{aligned}$